Erdős Problem 891

formal-conjectures · erdos-problems · ams-11

Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace
$p_1\cdots p_k$ with $p_1\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all
integers at most $p_1\cdots p_k$. By Dickson's conjecture [Wikipedia], there are infinitely many
$n'$ such that $\frac{L_k}{m}n'+1$ is prime for all $1\leq m < p_1\cdots p_k$. It follows that,
if $n=L_kn'+1$, then all integers in $[n,n+p_1\cdots p_k-1)$ have at most $k$ prime factors.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/891
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Dickson%27s_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/891.lean
Local target: Corpus.Erdos891.erdos_891.variants.weisenberg
Source SHA-256: 7f4e98ceecf3594f6b40397039bc885579f2be1f6815f132cd14af943b7be1ac
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record